Compound Interest

Sommaire

In short: Compound interest is money earned on both the starting deposit and on the interest that has piled up, which makes a balance climb faster than under simple interest.

Algebra 1Functions and GraphsF-IF.C.8b

What is Compound Interest?

Each time interest lands in your account, compound interest means the next round is computed on the larger balance, so interest begins earning interest of its own.

The formula is $A = P\left(1 + \frac{r}{n}\right)^{nt}$: where $P$ is the opening deposit, $r$ is the yearly rate in decimal form, $n$ counts how often interest posts in a year, and $t$ is the time in years. It is an exponential function whose factor per period is $1 + \frac{r}{n}$.

How to spot it

A savings account, loan or investment says interest is added monthly, quarterly or yearly.
The problem gives a principal, an annual rate and a number of years.
The word compounded appears with a frequency, a sign that compound interest rather than simple interest applies.
You are asked for the final balance rather than just the interest earned.

The method

Write down P, r as a decimal, n per year and t in years.
Compute the rate per period $\frac{r}{n}$ and the total number of periods $nt$.
Evaluate $\left(1 + \frac{r}{n}\right)^{nt}$ with a calculator, keeping extra decimals.
Multiply by P to get A, then subtract P if you want the interest earned.

$2,000 at 5% compounded annually: each year’s gain is larger than the last.

Compound Interest: solved example

Problem

Marcus puts $3,500 into a savings plan that pays 6% a year with interest posted every quarter. Find his balance three years later and the interest he collected.

Solution

Here $P = 3{,}500$, $r = 0.06$, $n = 4$ and $t = 3$. The rate per quarter is $\frac{0.06}{4} = 0.015$, over $nt = 12$ quarters.

$A = 3{,}500(1.015)^{12} \approx 3{,}500 \times 1.195618 \approx 4{,}184.66$. The balance is $4,184.66, so the interest earned is $4{,}184.66 – 3{,}500 = \mathbf{684.66}$ dollars.

Try it

Question 1

Find the balance on $1,000 at 4% compounded annually for 2 years.

Show the answer

$1{,}000(1.04)^2 = 1{,}000(1.0816) = 1{,}081.60$, so $1,081.60.

Question 2

Two accounts have the same principal and rate, but one compounds monthly and one yearly. Which ends with more?

Show the answer

The monthly one. More compounding periods mean interest is added to the balance sooner.

Compound Interest vs. Simple Interest

Simple interest ignores earlier gains and is figured only on the original principal, $I = Prt$, so it adds the same dollar amount every year. Compounding includes the interest already earned, so the yearly gain keeps rising. Over long periods, the gap between the two becomes large.

Simple Interest

Keep going

Lessons

Exponential Functions and Growth

Worksheets and quizzes

Related words

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